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Linear Nonhomogeneous Systems of Differential Equations with Constant Coefficients

A normal linear inhomogeneous system of n equations with constant coefficients can be written as

where t is the independent variable (often t is time), xi (t) are unknown functions which are continuous and differentiable on an interval [a, b] of the real number axis t, aij (i, j = 1, ..., n) are the constant coefficients, fi (t) are given functions of the independent variable t. We assume that the functions xi (t), fi (t) and the coefficients aij may take both real and complex values.

We introduce the following vectors:

and the square matrix

Then the system of equations can be written in a more compact matrix form as

For nonhomogeneous linear systems, as well as in the case of a linear homogeneous equation, the following important theorem is valid:

The general solution of the nonhomogeneous system is the sum of the general solution of the associated homogeneous system and a particular solution of the nonhomogeneous system:

Methods of solutions of the homogeneous systems are considered on other web-pages of this section. Therefore, below we focus primarily on how to find a particular solution.

Another important property of linear inhomogeneous systems is the principle of superposition, which is formulated as follows:

If is a solution of the system with the inhomogeneous part and is a solution of the same system with the inhomogeneous part then the vector function

is a solution of the system with the inhomogeneous part

The most common methods of solution of the nonhomogeneous systems are the method of elimination, the method of undetermined coefficients (in the case where the function is a vector quasi-polynomial), and the method of variation of parameters. Consider these methods in more detail.

Elimination Method

This method allows to reduce the normal nonhomogeneous system of equations to a single equation of th order. This method is useful for solving systems of order

Method of Undetermined Coefficients

The method of undetermined coefficients is well suited for solving systems of equations, the inhomogeneous part of which is a quasi-polynomial.

A real vector quasi-polynomial is a vector function of the form

where are given real numbers, and are vector polynomials of degree For example, a vector polynomial is written as

where are -dimensional vectors ( is the number of equations in the system).

In the case when the inhomogeneous part is a vector quasi-polynomial, a particular solution is also given by a vector quasi-polynomial, similar in structure to

For example, if the nonhomogeneous function is

a particular solution should be sought in the form

where in the non-resonance case, i.e. when the index in the exponential function does not coincide with an eigenvalue If the index coincides with an eigenvalue i.e. in the so-called resonance case, the value of is chosen to be equal to the greatest length of the Jordan chain for the eigenvalue In practice, can be taken as the algebraic multiplicity of

Similar rules for determining the degree of the polynomials are used for quasi-polynomials of kind

Here the resonance case occurs when the number coincides with a complex eigenvalue of the matrix

After the structure of a particular solution is chosen, the unknown vector coefficients are found by substituting the expression for in the original system and equating the coefficients of the terms with equal powers of on the left and right side of each equation.

Method of Variation of Constants

The method of variation of constants (Lagrange method) is the common method of solution in the case of an arbitrary right-hand side

Suppose that the general solution of the associated homogeneous system is found and represented as

where is a fundamental system of solutions, i.e. a matrix of size whose columns are formed by linearly independent solutions of the homogeneous system, and is the vector of arbitrary constant numbers

We replace the constants with unknown functions and substitute the function in the nonhomogeneous system of equations:

Since the Wronskian of the system is not equal to zero, then there exists the inverse matrix Multiplying the last equation on the left by we obtain:

where is an arbitrary constant vector.

Then the general solution of the nonhomogeneous system can be written as

We see that a particular solution of the nonhomogeneous equation is represented by the formula

Thus, the solution of the nonhomogeneous equation can be expressed in quadratures for any inhomogeneous term In many problems, the corresponding integrals can be calculated analytically. This allows us to express the solution of the nonhomogeneous system explicitly.

See solved problems on Page 2.

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